Problem: $\dfrac{dy}{dx}=7y$, and $y=1$ when $x=0$. Solve the equation. Choose 1 answer: Choose 1 answer: (Choice A) A $y=e^{7x}$ (Choice B) B $y=e^{x}$ (Choice C) C $y=7e^{x}$ (Choice D) D $y=7e^{7x}$
Solution: The general solution of equations of the form $\dfrac{dy}{dx}=ky$ is $y=C\cdot e^{kx}$ for some constant $C$. This can be found using separation of variables. In our case, $k=7$, so $y=C\cdot e^{7x}$. Let's use the fact that $y=1$ when $x=0$ to find $C$ : $\begin{aligned} y&=C\cdot e^{7x} \\\\ 1&=C\cdot e^{7\cdot 0} \gray{\text{Plug }x=0\text{ and }y=1} \\\\ 1&=C \end{aligned}$ In conclusion, $y=e^{7x}$.